Analysis and Computation of Multidimensional Linear Complexity of Periodic Arrays

TL;DR

This paper surveys and extends methods for analyzing the multidimensional linear complexity of periodic arrays, providing new bounds, comparisons, computations, conjectures, and open-source tools for applications in information security.

Contribution

It introduces new results on the linear complexity of multidimensional arrays, generalizes existing bounds, and offers open-source software for array construction and complexity computation.

Findings

Generalized bounds for multidimensional linear complexity

Computed complexities for arrays with non-relatively prime periods

Formulated conjectures on asymptotic behavior of array complexities

Abstract

Linear complexity is an important parameter for arrays that are used in applications related to information security. In this work we survey constructions of two and three dimensional arrays, and present new results on the multidimensional linear complexity of periodic arrays obtained using the definition and method proposed in \cite{ArCaGoMoOrRuTi,GoHoMoRu,MoHoRu}. The results include a generalization of a bound for the linear complexity, a comparison with the measure of complexity for multisequences, and computations of the complexity of arrays with periods that are not relatively prime for which the ``unfolding method'' does not work. Conjectures for exact formulas and the asymptotic behavior of the complexity of some array constructions are formulated. We also present open source software for constructing multidimensional arrays and for computing their multidimensional linear complexity.